Lecture 35M From Marion's Book

1
(No Transcript)
2
Lagranges Equations with Undetermined
Multipliers Marion, Section 7.5

• Holonomic Constraints are defined as those which
  can be expressed as algebraic relations among the
  coordinates.
• If a system has only holonomic constraints
• ? We can always find a proper set of generalized
  coordinates in which the equations of motion do
  not explicitly depend on constraints.
• Constraints which depend on both the coordinates
  the velocities are non-holonomic unless the
  constraint equations can be integrated to give
  relations among the coordinates.

3

• A constraint which depends on both the
  coordinates the velocities is of the form
  F(xai,xai,t) 0
• It is non-holonomic unless it can be integrated
  to give another function G(xai,t) 0
• As an example of this, consider a constraint
  relation of the form ?i Ai xi B 0 (i
  1,2,3) (1)
• (1) is non-integrable ( so the constraint is
  non-holonomic) unless Ai B have the special
  forms f f(xi,t)
• Ai (?f/?xi), B (?f/?t)
• In that case, (1) becomes ?i(?f/?xi)(dxi/dt)
  (?f/?t) 0
• Or, (1) becomes (df/dt) 0, which can be
  integrated to find f(xi,t) const, or f(xi,t)
  - const 0, a holonomic constraint!

4

• This example shows that ? Constraints which can
  be written in the differential form
• ?j(?f/?qj)dqj (?f/?t)dt 0 (2)
• are holonomic constraints.
• Often, in practical physical situations, the
  constraints can be written in the differential
  form (2). If this is the case, the constraints
  can be explicitly incorporated into Lagranges
  Equations using the method of Lagranges
  undetermined multipliers (Sect. 6.6).

5

• ? The formalism of Sect. 6.6 (changing to the
  notation of Ch. 7) shows that if the holonomic
  constraints can be written in the differential
  form
• ?j(?fk/?qj)dqj 0 j 1,2, ..,s k 1,2,..m
  (A)
• (also can be shown explicitly using the
  variational form of Hamiltons Principle)
  Lagranges Equations become
• (?L/?qj) - (d/dt)?L/?qj ?k?k(t)(?fk/?qj)
  0 (B)
• ?k(t) ? Lagranges undetermined multipliers
• We could also have added a term (?fk/?t)dt to
  (A) still have gotten (B).

6

• Summary
• Lagranges Equations with constraints
• (?L/?qj) - (d/dt)?L/?qj ?k?k(t)(?fk/?qj) 0
• ?k(t) ? Lagranges undetermined multipliers
• The ?k(t) are determined as part of solution to
  the problem!
• The physical interpretation of the ?k(t) will be
  discussed next.

7

• Lagranges Equations
• (?L/?qj) - (d/dt)?L/?qj ?k?k(t)(?fk/?qj) 0
  (C)
• The major advantage to Lagrangian Mechanics
  Explicit inclusion of forces is not necessary.
  Emphasis is placed on the dynamics of the system
  rather than on the calculation of forces.
• Often, however, we might want or need to know the
  forces of constraint. It is explicitly shown in
  graduate texts ( well see in some examples)
  that the Lagrange multipliers ?k(t) can be used
  to calculate the (generalized) forces of
  constraint!

8
Generalized Forces

• Specifically, it can be shown that the
  Generalized Forces of Constraint Qj
  (corresponding to the generalized coordinates qj)
  are given by
• Qj ? ?k?k(t)(?fk/?qj) The last term
  in (C)!
• Just as the generalized coordinates qj do not
  necessarily have units of length, the
  corresponding generalized forces Qj do not
  necessarily have units of force! Well see this
  in the examples!

9
Example 7.9

• A disk, radius R, rolls without slipping down an
  inclined plane (total length ?) as shown. Find
  the equations of motion, the force of constraint,
  the angular acceleration.
• Note The generalized coordinates y ? are not
  independent.
• They are related by y R?.
• ? The constraint equation is
• f(y,?) y - R? 0
• This is equivalent to the
• differential versions
• (?f/?y) 1 (?f/??) -R
• Worked on the board with 2 separate methods
  without with Lagrange multipliers.

10
Example 7.10

• A particle of mass m starts from rest on top of
  smooth hemisphere of radius a (figure). Find the
  force of constraint determine the angle at
  which m leaves hemisphere.

11

• Summary
• Lagranges Equations with constraints
• (?L/?qj) - (d/dt)?L/?qj ?k?k(t)(?fk/?qj)
  0 (C)
• The usefulness of Lagrange Eqtns with
  undetermined multipliers
• The Lagrange multipliers ?k(t) can be used to
  obtain the forces of constraint. These are often
  needed. Get ?k(t) as part of the solution to the
  equations of motion.
• When a proper set of generalized coordinates is
  not wanted or is too difficult to get, we can use
  this method to increase the number of generalized
  coordinates by including the constraints
  explicitly in the equations of motion.